the graph of which function will have a maximum and a $y$-intercept of 4?\n\n$f(x) = 4x^2 + 6x - 1$\n$f(x) =…

the graph of which function will have a maximum and a $y$-intercept of 4?\n\n$f(x) = 4x^2 + 6x - 1$\n$f(x) = -4x^2 + 8x + 5$\n$f(x) = -x^2 + 2x + 4$\n$f(x) = x^2 + 4x - 4$

the graph of which function will have a maximum and a $y$-intercept of 4?\n\n$f(x) = 4x^2 + 6x - 1$\n$f(x) = -4x^2 + 8x + 5$\n$f(x) = -x^2 + 2x + 4$\n$f(x) = x^2 + 4x - 4$

Answer

Explanation:

Step1: Identify condition for a maximum

A quadratic function $f(x) = ax^2 + bx + c$ has a maximum if the leading coefficient $a$ is negative ($a < 0$).

Step2: Identify condition for y-intercept

The $y$-intercept occurs at $f(0)$. For $f(x) = ax^2 + bx + c$, the $y$-intercept is the constant term $c$. We need $c = 4$.

Step3: Evaluate the given options

  1. $f(x) = 4x^2 + 6x - 1$: $a=4$ (minimum), $c=-1$.
  2. $f(x) = -4x^2 + 8x + 5$: $a=-4$ (maximum), $c=5$.
  3. $f(x) = -x^2 + 2x + 4$: $a=-1$ (maximum), $c=4$.
  4. $f(x) = x^2 + 4x - 4$: $a=1$ (minimum), $c=-4$.

Answer:

$f(x) = -x^2 + 2x + 4$